Aldous' Spectral Gap Conjecture for Normal Sets

TL;DR

This paper extends Aldous' spectral gap conjecture to normal sets in the symmetric group, showing eigenvalue similarities for conjugacy classes and discussing limitations and conjectures for general symmetric sets.

Contribution

It demonstrates spectral gap similarities for conjugacy classes in $S_n$ and explores the limitations for arbitrary normal sets, proposing a new conjecture for symmetric sets.

Findings

Eigenvalues for conjugacy classes match those of certain Schreier graphs.

The spectral gap similarity does not hold for all normal sets.

A new conjecture is proposed for arbitrary symmetric sets.

Abstract

Let $S_n$ denote the symmetric group on $n$ elements, and $\Sigma\subseteq S_{n}$ a symmetric subset of permutations. Aldous' spectral gap conjecture, proved by Caputo, Liggett and Richthammer [arXiv:0906.1238], states that if $\Sigma$ is a set of transpositions, then the second eigenvalue of the Cayley graph $\mathrm{Cay}\left(S_{n},\Sigma\right)$ is identical to the second eigenvalue of the Schreier graph on $n$ vertices depicting the action of $S_{n}$ on $\left\{ 1,\ldots,n\right\}$. Inspired by this seminal result, we study similar questions for other types of sets in $S_{n}$. Specifically, we consider normal sets: sets that are invariant under conjugation. Relying on character bounds due to Larsen and Shalev [2008], we show that for large enough $n$, if $\Sigma\subset S_{n}$ is a full conjugacy class, then the second eigenvalue of $\mathrm{Cay}\left(S_{n},\Sigma\right)$ is roughly identical to the second eigenvalue of the Schreier graph depicting the action of $S_{n}$ on ordered $4$-tuples of elements from $\left\{ 1,\ldots,n\right\}$. We further show that this type of result does not hold when $\Sigma$ is an arbitrary normal set, but a slightly weaker one does hold. We state a conjecture in the same spirit regarding an arbitrary symmetric set $\Sigma\subset S_{n}$, which yields surprisingly strong consequences.